Coulomb Rydberg electron <Emphasis Type="Italic">L</Emphasis>-mixing in collisions with atomic ions

The cross sections of the Rydberg electron L-mixing in a hydrogen atom and a hydrogen-like ion are calculated for slow collisions with atomic ions H*(n, L) + A⁺ = H*(n, L′) + A⁺ without variation of the principal quantum number n. The probability of the L-mixing L → L′ is associated with the quantum interference of the wave functions of adiabatic states, i.e., with the mixing of the time phases of these functions exp(?i∫E _k (t)dt). The effective cross section of such L-mixing for the states with n = 28 are 4–5 orders of magnitude greater than the cross sections determined in previous investigations. The expansion coefficients of spherical Coulomb wave functions in terms of parabolic ones and vice versa, which are necessary for determining cross sections, are calculated on the basis of a comprehensive analysis of the spatial properties of these functions.     

The real solution to scalar field equation in 5D black string space

After the nontrivial quantum parameters Ω _n and quantum potentials V _n obtained in our previous research, the circumstance of a real scalar wave in the bulk is studied with the similar method of Brevik and Simonsen (Gen. Rel. Grav. 33:1839, 2001). The equation of a massless scalar field is solved numerically under the boundary conditions near the inner horizon r _e and the outer horizon r _c . Unlike the usual wave function Ψ_ωl in 4D, quantum number n introduces a new functions Ψ_{ωl n} , whose potentials are higher and wider with bigger n. Using the tangent approximation, a full boundary value problem about the Schrödinger-like equation is solved. With a convenient replacement of the 5D continuous potential by square barrier, the reflection and transmission coefficients are obtained. If extra dimension does exist and is visible at the neighborhood of black holes, the unique wave function Ψ_{ωl n} may say something to it.     

Intuitionistic Quantum Logic of an <Emphasis Type="Italic">n</Emphasis>-level System

A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*-algebra M _n (?) of complex n×n matrices. This leads to an explicit expression for the pointfree quantum phase space Σ _n and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen–Specker Theorem.In our approach, the nondistributive lattice ?(M _n (?)) of projections in M _n (?) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice \(\mathcal{O}(\Sigma_{n})\) of functions from the poset \(\mathcal{C}(M_{n}(\mathbb{C}))\) of all unital commutative C*-subalgebras C of M _n (?) to ?(M _n (?)). The lattice \(\mathcal{O}(\Sigma_{n})\) is essentially the (pointfree) topology of the quantum phase space Σ _n , and as such defines a Heyting algebra. Each element of \(\mathcal{O}(\Sigma_{n})\) corresponds to a “Bohrified” proposition, in the sense that to each classical context \(C\in\mathcal{C}(M_{n}(\mathbb{C}))\) it associates a yes-no question (i.e. an element of the Boolean lattice ?(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann.     

Scattering of Josephson plasmons on random quantum jumpers in a disordered <Emphasis Type="Italic">I</Emphasis>-layer of an <Emphasis Type="Italic">S</Emphasis>–<Emphasis Type="Italic">I</Emphasis>–<Emphasis Type="Italic">S</Emphasis> junction

A formula for the relaxation time of Josephson plasmons on random quantum jumpers, i.e., quantum resonant-percolation trajectories (QRPT) in a disordered I-layer of a tunnel S–I–S junction is derived. Domain Ω_r (μ ? E₀, c), in which the strongest plasmon damping takes place, is plotted in the plane of parameters (μ ? E₀, c).     

Quantum groups as generalized gauge symmetries in WZNW models. Part II. The quantized model

This is the second part of a paper dealing with the “internal” (gauge) symmetry of the Wess–Zumino–Novikov–Witten (WZNW) model on a compact Lie group G. It contains a systematic exposition, for G = SU(n), of the canonical quantization based on the study of the classical model (performed in the first part) following the quantum group symmetric approach first advocated by L.D. Faddeev and collaborators. The internal symmetry of the quantized model is carried by the chiral WZNW zero modes satisfying quadratic exchange relations and an n-linear determinant condition. For generic values of the deformation parameter the Fock representation of the zero modes’ algebra gives rise to a model space of U _q (sl(n)). The relevant root of unity case is studied in detail for n = 2 when a “restricted” (finite dimensional) quotient quantum group is shown to appear in a natural way. The module structure of the zero modes’ Fock space provides a specific duality with the solutions of the Knizhnik–Zamolodchikov equation for the four point functions of primary fields suggesting the existence of an extended state space of logarithmic CFT type. Combining left and right zero modes (i.e., returning to the 2D model), the rational CFT structure shows up in a setting reminiscent to covariant quantization of gauge theories in which the restricted quantum group plays the role of a generalized gauge symmetry.     

Coulomb Problem for <Emphasis Type="Italic">Z</Emphasis> &gt; Z<Subscript>cr</Subscript> in Doped Graphene

The dynamics of charge carriers in doped graphene, i.e., graphene with a gap in the energy spectrum depending on the substrate, in the presence of a Coulomb impurity with charge Z is considered within the effective two-dimensional Dirac equation. The wave functions of carriers with conserved angular momentum J = M + 1/2 are determined for a Coulomb potential modified at small distances. This case, just as any two-dimensional physical system, admits both integer and half-integer quantization of the orbital angular momentum in plane, M = 0, ±1, ±2, …. For J = 0, ±1/2, ±1, critical values of the effective charge Z_cr(J, n) are calculated for which a level with angular momentum J and radial quantum numbers n = 0 and n = 1 reaches the upper boundary of the valence band. For Z < Z_cr (J, n = 0), the energy of a level is presented as a function of charge Z for the lowest values of orbital angular momentum M, the level with J = 0 being the first to descend to the band edge. For Z>Z_cr (J, n = 0), scattering phases are calculated as a function of hole energy for several values of supercriticality, as well as the positions ε₀ and widths γ of quasistationary states as a function of supercriticality. The values of ε₀* and width γ* are pointed out for which quasidiscrete levels may show up as Breit–Wigner resonances in the scattering of holes by a supercritical impurity. Since the phases are real, the partial scattering matrix is unitary, so that the radial Dirac equation is consistent even for Z > Z_cr. In this single-particle approximation, there is no spontaneous creation of electron–hole pairs, and the impurity charge cannot be screened by this mechanism.     

Bad communities with high modularity

In this paper we discuss some problematic aspects of Newman and Girvan’s modularity function Q _N . Given a graph G, the modularity of G can be written as Q _N = Q _f ? Q ₀, where Q _f is the intracluster edge fraction of G and Q ₀ is the expected intracluster edge fraction of the null model, i.e., a randomly connected graph with same expected degree distribution as G. It follows that the maximization of Q _N must accomodate two factors pulling in opposite directions:Q _f favors a small number of clusters and Q ₀ favors many balanced (i.e., with approximately equal degrees) clusters. In certain cases the Q ₀ term can cause overestimation of the true cluster number; this is the opposite of the well-known underestimation effect caused by the “resolution limit” of modularity. We illustrate the overestimation effect by constructing families of graphs with a “natural” community structure which, however, does not maximize modularity. In fact, we show there exist graphs G with a “natural clustering” V of G and another, balanced clustering U of G such that (i) the pair (G, U) has higher modularity than (G, V) and (ii) V and U are arbitrarily different.     

Fluctuations and a rigorous uncertainty relation of trigonometric operators of the phase and the number of photons of an electromagnetic field for general quantum superpositions of coherent states

The quantum-statistical properties of states of an electromagnetic field of general superpositions of coherent states of the form of N _α,β(α?+e ^iξ β? are investigated. Formulas for the fluctuations (variances) of Hermitian trigonometric phase field operators ? ≡ côs φ, ? ≡ sîn φ (the so-called “Susskind–Glogower operators”) are found. Expressions for the rigorous uncertainty relations (Cauchy inequalities) for operators of the number of photons and trigonometric phase operators, as well as for operators ? and ?, are found and analyzed. The states of amplitude \({N_{\alpha ,\beta }}\left( {\left| {{{\sqrt {ne} }^{i\varphi }}\rangle + {e^{i\xi }}\left| {{{\sqrt {{n_\beta }e} }^{i\varphi }}\rangle } \right.} \right.} \right)\), φ = φ_α = φ_β, and phase \({N_{\alpha ,\beta }}\left( {\left| {{{\sqrt {ne} }^{i{\varphi _\alpha }}}\rangle + {e^{i\xi }}\left| {{{\sqrt {ne} }^{i{\varphi _\beta }}}\rangle } \right.} \right.} \right)\), n = n _α = n _β, superpositions of coherent states are considered separately. The types of quantum superpositions of meso- and macroscales (n _α, n _β » 1) are found for which the sines and/or cosines of the phase of the field can be measured accurately, since, under certain conditions, the quantum fluctuations of these quantities are close to zero. A simultaneous accurate measurement of cosφ and sinφ is possible for amplitude superpositions, while an accurate measurement of one of these trigonometric phase functions is possible in the case of certain phase superpositions. Amplitude superpositions of coherent states with a vacuum state are quantum states of the field with a “maximum” level of the quantum uncertainty both in the case of a mesoscopic scale and in the case of a macroscopic scale of the field with an average number of photons n _α/β ≈ 0, n _β/α » 1.     

Path integral approach for spaces of nonconstant curvature in three dimensions

In this contribution, I show that it is possible to construct three-dimensional spaces of nonconstant curvature, i.e., three-dimensional Darboux spaces. Two-dimensional Darboux spaces have been introduced by Kalnins et al., with a path integral approach by the present author. In comparison to two dimensions, in three dimensions it is necessary to add a curvature term in the Lagrangian in order that the quantum motion can be properly defined. Once this is done, it turns out that, in the two three-dimensional Darboux spaces which are discussed in this paper, the quantum motion is similar to the two-dimensional case. In D _3d-I, we find seven coordinate systems which separate the Schrödinger equation. For the second space, D _3d-II, all coordinate systems of flat three-dimensional Euclidean space which separate the Schrödinger equation also separate the Schrödinger equation in D _3d-II. I solve the path integral on D _3d-I in the (u, v, w) system and on D _3d-II in the (u, v, w) system and in spherical coordinates.     

New insights into quantum and classical correlations in XY spin models

We compute quantum dissonance Q (non-entangled quantum correlation), entanglement E, quantum discord D (total quantum correlation) and classical correlation C for spin pairs at any distance in the infinite XY spin-1/2 chains, i.e., the anisotropic XY model and the isotropic XY model with three-spin interactions. We obtain two simple dominance relations: C ≥ E and D ≥ E + Q Except this, there are no other simple ordering relations between them. We also show that Q can detect the special points of the system where the entanglement just appears or completely disappears. In addition, it is worthwhile to mention that dissonance and classical correlation can also clearly spotlight the critical points of quantum phase transitions in XY spin-1/2 chains.     

A More Efficient Contextuality Distillation Protocol

Based on the fact that both nonlocality and contextuality are resource theories, it is natural to ask how to amplify them more efficiently. In this paper, we present a contextuality distillation protocol which produces an n-cycle box B ? B ^′ from two given n-cycle boxes B and B ^′. It works efficiently for a class of contextual n-cycle (n ≥?4) boxes which we termed as “the generalized correlated contextual n-cycle boxes”. For any two generalized correlated contextual n-cycle boxes B and B ^′, B ? B ^′ is more contextual than both B and B ^′. Moreover, they can be distilled toward to the maximally contextual box C H _n as the times of iteration goes to infinity. Among the known protocols, our protocol has the strongest approximate ability and is optimal in terms of its distillation rate. What is worth noting is that our protocol can witness a larger set of nonlocal boxes that make communication complexity trivial than the protocol in Brunner and Skrzypczyk (Phys. Rev. Lett. 102, 160403 2009), this might be helpful for exploring the problem that why quantum nonlocality is limited.     

Decoherence control mechanisms of a charged magneto-oscillator in contact with different environments

In this work, we consider two different techniques based on reservoir engineering process and quantum Zeno control method to analyze the decoherence control mechanism of a charged magneto-oscillator in contact with different type of environment. Our analysis reveals that both the control mechanisms are very much sensitive on the details of different environmental spectrum (J?(ω)), and also on different system and reservoir parameters, e.g., external magnetic field (r _c ), confinement length (r ₀), temperature (T), cut-off frequency of reservoir spectrum (ω _cut ), and measurement interval (τ). We also demonstrate the manipulation scheme of the continuous passage from decay suppression to decay acceleration by tuning the above mentioned system or reservoir parameters, e.g., r _c , r ₀, T and τ.     

Composite-Particles (Boson,Fermion) Theory of Fractional Quantum Hall Effect

A theory is developed for fractional quantum Hall effect in terms of composite (c)-bosons (fermions) without useing Laughlin’s results about the fractional charge. Here the c-particle (fermion, boson) is defined as a bound composite fermion (boson) containing a conduction electron and an even (odd) number of fluxons (elementary magnetic fluxes). The Bose-condensed c-bosons, each containing an electron and an odd number m of fluxons at the filling factor ν=1/m is shown to generate the Hall conductivity plateau value m e ²/h, where the density of c-particles, \(n_{\phi }^{(m)}\), either bosonic or fermionic, with m fluxons is given by \(n_{\phi }^{(m)}=n_{\mathrm {e}}/m\), n _e = electron density. The only assumption is that any c-fermion carries a charge magnitude equal to the electron charge e. The quantum Hall state is shown to be more stable at ν=1/3 than at ν=1.     

Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics

The Rényi entropies R_p [ ρ], p> 0,≠ 1 of the highly-excited quantum states of the D-dimensional isotropicharmonic oscillator are analytically determined by use of the strong asymptotics of theorthogonal polynomials which control the wavefunctions of these states, the Laguerrepolynomials. This Rydberg energetic region is where the transition from classical toquantum correspondence takes place. We first realize that these entropies are closelyconnected to the entropic moments of the quantum-mechanical probability ρ_n(r)density of the Rydberg wavefunctions Ψ_{n,l, { μ}}(r); so, to the?_p-norms of the associated Laguerrepolynomials. Then, we determine the asymptotics n → ∞ of these norms by use of modern techniques ofapproximation theory based on the strong Laguerre asymptotics. Finally, we determine thedominant term of the Rényi entropies of the Rydberg states explicitly in terms of thehyperquantum numbers (n,l), the parameter order p and the universedimensionality D for all possible cases D ≥ 1. We find that (a) theRényi entropy power decreases monotonically as the order p is increasing and (b) thedisequilibrium (closely related to the second order Rényi entropy), which quantifies theseparation of the electron distribution from equiprobability, has a quasi-Gaussianbehavior in terms of D.     

A New Family of Solvable Pearson-Dirichlet Random Walks

An n-step Pearson-Gamma random walk in ? ^d starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations. The gamma distribution of each step length has a shape parameter q>0. Constrained random walks of n steps in ? ^d are obtained from the latter walks by imposing that the sum of the step lengths is equal to a fixed value. Simple closed-form expressions were obtained in particular for the distribution of the endpoint of such constrained walks for any d≥d ₀ and any n≥2 when q is either \(q = \frac{d}{2} - 1 \) (d ₀=3) or q=d?1 (d ₀=2) (Le Caër in J. Stat. Phys. 140:728–751, 2010). When the total walk length is chosen, without loss of generality, to be equal to 1, then the constrained step lengths have a Dirichlet distribution whose parameters are all equal to q and the associated walk is thus named a Pearson-Dirichlet random walk. The density of the endpoint position of a n-step planar walk of this type (n≥2), with q=d=2, was shown recently to be a weighted mixture of 1+floor(n/2) endpoint densities of planar Pearson-Dirichlet walks with q=1 (Beghin and Orsingher in Stochastics 82:201–229, 2010). The previous result is generalized to any walk space dimension and any number of steps n≥2 when the parameter of the Pearson-Dirichlet random walk is q=d>1. We rely on the connection between an unconstrained random walk and a constrained one, which have both the same n and the same q=d, to obtain a closed-form expression of the endpoint density. The latter is a weighted mixture of 1+floor(n/2) densities with simple forms, equivalently expressed as a product of a power and a Gauss hypergeometric function. The weights are products of factors which depends both on d and n and Bessel numbers independent of d.     

Dynamical Characteristics of a Non-canonical Scalar-Torsion Model of Dark Energy

In this paper, we analyze the phase-space of a model of dark energy in which a non-canonical scalar field (tachyon) non-minimally coupled to torsion scalar in the framework of teleparallelism. Scalar field potential and non-minimal coupling function are chosen as V(?) = V₀?ⁿ and f(?) = ?^N, respectively. We obtain a critical point that behaves like a stable or saddle point depending on the values of N and n. Additionally we find an unstable critical line. We have shown such a behavior of critical points using numerical computations and phase-space trajectories explicitly.     

The effect of non-linear capacitances in the localization properties of aperiodic dual electric transmission lines

This study investigates the localization properties of dual electric transmission lines with non-linear capacitances. The V_C,n voltage across each capacitor is selected as a non-linear function of the electric charge q_n, i.e., V_C,n = q_n(1/C_n -ε_n|q_n|²)where C_n is the linear part of the capacitance and ε_n the amplitude of the non-linear term. We follow a binary distribution of values of ε_n, according to the Thue-Morse m-tupling sequence. The localization behavior of this non-linear case indicates that the case m = 2 does not belong to the m ≥ 3, family because when m changes from m = 2 to m = 3, the number of extended states diminishes dramatically. This proves the topological difference of the m = 2 and m = 3 families. However, by increasing m values, localization behavior of the m-tupling family resembles that of the m = 2, case because the system begins to regain its extended states. The exact same result was obtained recently in the study of linear direct transmission lines with m-tupling distribution of inductances. Consequently, we state that the localization behavior of the m-tupling family as a function of the m value is independent of both the linear and the non-linear system under study, but independent of the kind of transmission line (dual or direct). This is curious behavior of the m-tupling family and thus deserves more scholarly attention.